# Chapter 4: Measuring motion

## 4.2 Velocity

Velocity is the rate of change of displacement: velocity tells us how an object’s position is changing in time.

$\text{velocity} = \frac{\text{change in position}}{\text{change in time}}$

Or, in symbols (for motion along only the $$x$$ axis):

$v_x = \frac{\Delta x}{\Delta t} \tag{4.1}$

The symbol $$\Delta$$ is a capital Greek letter delta, and it always refers to a change. Mathematically, delta refers to the difference between final and initial value of a particular quantity. For example $$\Delta x$$ is the change in the $$x$$ coordinate of an object’s position, and is given by

$\Delta x = x_f – x_i \tag{4.2}$

Note how we use subscripts to be a bit more specific in what we’re referring to with each variable. In the equation defining $$v$$ we used the subscript $$x$$ to indicate velocity along the $$x$$ axis, and in equation defining $$\Delta$$ we used subscripts $$f$$ and $$i$$ to indicate the final and initial values of the variable $$x$$.

It is common practice to say $$t_i = 0$$ and $$t_f = t$$. This means we often simply use $$t$$ instead of $$\Delta t$$. This reflects the practice that $$t$$ usually refers to some elapsed time, such as you would measure with a stopwatch.

Because position is a vector, velocity is also a vector. The direction of the velocity vector is the direction of motion. When working with one-dimensional motion, a positive or negative sign indicates direction; for two-dimensional motion we’ll use vector notation. For two-dimensional motion, we use the rate of change of each component of the position:

$\vec{v} = \underbrace{\left(\frac{\Delta x}{\Delta t}\right)}_{v_x}\hat{x} + \underbrace{\left(\frac{\Delta y}{\Delta t}\right)}_{v_y}\hat{y} \tag{4.3}$

The magnitude of an object’s velocity is its speed:

$v = \sqrt{\left(v_x\right)^2 + \left(v_y\right)^2} = \text{speed}$

(Recall from chapter 2 that the symbol $$v$$ represents the magnitude of vector $$\vec{v}$$.)

The SI unit for velocity (and speed) is meters per second (m/s).

#### Example 4.2

A cat is asleep, when a bad dream suddenly wakes it up and it runs across the room. The cat wakes up at the time 1:42:03, and reaches the other side of the room at 1:42:11. The position of the box the cat was sleeping in is

$\vec{r}_1 = x_1\hat{x} + y_1\hat{y} = 12\hat{x} – 4\hat{y}\ \textrm{m}$

and the position the cat ends at is

$\vec{r}_2 = x_2\hat{x} + y_2\hat{y} = 2\hat{x} + 8\hat{y}\ \textrm{m}$

What were the cat’s velocity and speed?

Velocity is the rate of change of position. With 2D motion, we need to consider both vector components.

\begin{align*} \vec{v} &= \frac{\Delta \vec{r}}{\Delta t} \\ &= \frac{\Delta x}{\Delta t}\hat{x} + \frac{\Delta y}{\Delta t}\hat{y} \\ &= \frac{x_2 – x_1}{\Delta t}\hat{x} + \frac{y_2 – y_1}{\Delta t}\hat{y} \\ &= \frac{2 – 12\ \textrm{m}}{8\ \textrm{s}}\hat{x} + \frac{8 – (-4)\ \textrm{m}}{8\ \textrm{s}}\hat{y} \\ &= -1.25\hat{x} + 1.5\hat{y}\ \textrm{m/s} \end{align*}

Note that the cat’s velocity is a vector, given here in component form. The cat’s speed is the magnitude of its velocity:

\begin{align*} v &= \sqrt{v_x^2 + v_y^2} \\ &= \sqrt{(-1.25\ \textrm{m/s})^2 + (1.5\ \textrm{m/s})^2} \\ &= 1.95\ \textrm{m/s} \end{align*}